Suppose that a density is of the form $(k + 1)x^{k}$ for some constant $k>1$ and $0 \leq x \leq 1$.
What is the mean associated with this density?
So far, I have established that $E[X] = \int_{0}^{1}x(k+1)x^{k}dx$, so I would need to find the antiderivate to get the CDF, and subtract the CDF evaluated at $x = 1$ by the CDF evaluated at $x = 0$.
Where I am having trouble is finding the antiderivative for this equation. I believe part of this antiderivative is comprised of $x^{k+1}$, because then that would account for the $(k+1)$ coefficient, and would allow $x$ to be raised to the power of $k$. But this does not account for the $x$ term in the defintion of the expected value.